49 citations · 70 across the 11 of their papers we have counts for
14 papers
Concentration of mass of solutions to aggregation-diffusion equations
Grzegorz Karch, Krzysztof Krawczyk, Alexandre Boritchev
We consider the aggregation-diffusion equation in the whole space with a mildly singular interaction kernel K = K(x) which behaves like |x|^k near the origin for some k (0, 2…
Sharp well-posedness and blowup results for parabolic systems of the Keller-Segel type
Piotr Biler, Alexandre Boritchev, Lorenzo Brandolese
We study two toy models obtained after a slight modification of the nonlinearity of the usual doubly parabolic Keller-Segel system. For these toy models, both consisting of a syste…
Large global solutions of the parabolic-parabolic Keller-Segel system in higher dimensions
Piotr Biler, Alexandre Boritchev, Lorenzo Brandolese
We study the global existence of the parabolic-parabolic Keller-Segel system in . We prove that initial data of arbitrary size give rise to global solutions provide…
Sharp Sobolev estimates for concentration of solutions to an aggregation-diffusion equation
Piotr Biler, Alexandre Boritchev, Grzegorz Karch +1
We consider the drift-diffusion equation in the whole space with global-in-time solutions bounded in all Sobolev spaces; for simplicity, we…
Concentration phenomena in a diffusive aggregation model
Piotr Biler, Alexandre Boritchev, Grzegorz Karch +1
We consider the drift-diffusion equation in the whole space with global-in-time bounded solutions. Mass concentration pheno…
Intermittency of Riemann's non-differentiable function through the fourth-order flatness
Alexandre Boritchev, Daniel Eceizabarrena, Victor Vilaça da Rocha
Riemann's non-differentiable function is one of the most famous examples of continuous but nowhere differentiable functions, but it has also been shown to be relevant from a physic…
Decaying turbulence for the fractional subcritical Burgers equation
Alexandre Boritchev
We consider the fractional unforced Burgers equation in the one-dimensional space-periodic setting: $$\partial u/\partial t+(f(u))_x +νΛ^α u= 0, t \geq 0,\ \mathbb{x} \in \mathbb{T…
Exponential convergence to the stationary measure for a class of 1D Lagrangian systems with random forcing
Alexandre Boritchev
We prove exponential convergence to the stationary measure for a class of 1d Lagrangian systems with random forcing in the space-periodic setting: $$ ϕ_t+ϕ_x^2/2=F^ω, x \in S^1 = \…
Multidimensional potential Burgers turbulence
Alexandre Boritchev
We consider the multidimensional generalised stochastic Burgers equation in the space-periodic setting: $\m…
Turbulence for the generalised Burgers equation
Alexandre Boritchev
In this survey, we review the results on turbulence for the generalised Burgers equation on the circle: u_t+f'(u)u_x=νu_{xx}+η,\ x \in S^1=\R/\Z, obtained by A.Biryuk and the autho…
Decaying Turbulence in Generalised Burgers Equation
Alexandre Boritchev
We consider the generalised Burgers equation …
On hyperbolicity of minimizers for 1D random Lagrangian systems
Alexandre Boritchev, Konstantin Khanin
We prove hyperbolicity of global minimizers for random Lagrangian systems in dimension 1. The proof considerably simplifies a related result in [2]. The conditions for hyperbolicit…
Sharp Estimates for Turbulence in White-Forced Generalised Burgers Equation
Alexandre Boritchev
We consider the non-homogeneous generalised Burgers equation \frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - ν\frac{\partial^2 u}{\partial x^2} = η,\ t \geq 0,…
Estimates for Solutions of a Low-Viscosity Kick-Forced Generalised Burgers Equation
Alexandre Boritchev
We consider a non-homogeneous generalised Burgers equation: $$ \frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - ν\frac{\partial^2 u}{\partial x^2} = η^ω,\quad t…